Theory of multidimensional Delsarte – Lions transmutation operators. II - Blackmore, D.; Prykarpatsky, A. K.; Prykarpatsky, Ya. A.; Samoilenko, A. M. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
UDC 517.9 The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand – Levitan – Marchenko equations that describe these operators are studied by using sutable differential de Rham – Hodge – Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang – Mills equations. Soliton solutions of a certain class of dynamical systems are discussed.